Consider the embedding of the $\mathbb{S}^2$ in $\mathbb{R}^3$,
\begin{align}
x & = r \sin \theta \cos \phi,\tag1 \\
y & = r \sin \theta \sin \phi,\tag2 \\
z & = r \cos \theta,\tag3
\end{align}
with $r$ fixed at $r=1$.

It has $SO(3)$ symmetry. 
So now I want to express the generators of $SO(3)$, \begin{equation}
L^{X^iX^j} = X^i \frac{\partial}{\partial X^j} - X^j \frac{\partial}{\partial X^i}\tag4
\end{equation} in the terms of the the intrinsic co-ordinates $(r, \theta, \phi)$.

**Method 1** ( Similar to what we do when finding the angular momentum operators in spherical polar co-ordinates but Which according the answers and comments [here](https://math.stackexchange.com/questions/4997446/different-expressions-for-frac-partial-partial-z-in-spherical-polar-coord?noredirect=1#comment10711786_4997589) is a miscalculation):

Consider $L^{YZ}=y\frac{\partial}{\partial z} - z \frac{\partial}{\partial y}$ (Rotation in Y,Z co-ordinates) 

we express $\partial_z$ and $\partial_y$ in terms of $\partial_\theta$, $\partial_\phi$, using equations (1), (2) and (3) and the chain rule and plug in the equation for the generator. We first treat r as a variable. 
\begin{equation}
\frac{\partial}{\partial z} = \cos \theta \, \frac{\partial}{\partial r} - \frac{\sin \theta}{r} \, \frac{\partial}{\partial \theta}.
\end{equation}

\begin{equation}
\frac{\partial}{\partial y} = \sin \theta \sin \phi \, \frac{\partial}{\partial r} + \frac{\cos \theta \sin \phi}{r} \, \frac{\partial}{\partial \theta} + \frac{\cos \phi}{r \sin \theta} \, \frac{\partial}{\partial \phi}.
\end{equation}

with r fixed at $r=1$,
we have,
\begin{align}
\frac{\partial}{\partial z} &= - \sin \theta\, \frac{\partial}{\partial \theta},\tag5 \\
\frac{\partial}{\partial y} &= \cos \theta \sin \phi \, \frac{\partial}{\partial \theta} + \frac{\cos \phi}{\sin \theta} \, \frac{\partial}{\partial \phi}. \tag6
\end{align}
Gives the correct answer,
\begin{equation}L^{YZ}=\cot \theta \cos \phi \, \frac{\partial}{\partial \phi} + \sin \phi \, \frac{\partial}{\partial \theta}\end{equation}.

**Method 2**:
Working with covectors and the going to dual basis. **Please check the answer [here](https://math.stackexchange.com/a/4997600/995988)** We get the expressions for the dual basis,
\begin{equation}
\begin{bmatrix} 
\frac{\partial}{\partial y} \\ 
\frac{\partial}{\partial z} 
\end{bmatrix} 
= 
\begin{bmatrix} 
0 & \csc\theta\sec\phi \\ 
-\csc\theta & \cot\theta\csc\theta\tan\phi 
\end{bmatrix} 
\begin{bmatrix} 
\frac{\partial}{\partial\theta} \\ 
\frac{\partial}{\partial\phi} 
\end{bmatrix}\tag7
\end{equation}
We get the correct expression for $L^{YZ}$.

Method 3 (My [attempt](https://math.stackexchange.com/a/4997589/995988) a the justification):

Consider the map $f: (x, y, z) \rightarrow (\theta, \phi)$ defined by:
\begin{align}
\theta &= \cos^{-1}(z), \\
\phi &= \tan^{-1}\left(\frac{y}{x}\right).
\end{align}
The Jacobian matrix of this transformation is given by:
\begin{align}
J = 
\begin{pmatrix}
\frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y} & \frac{\partial \theta}{\partial z} \\
\frac{\partial \phi}{\partial x} & \frac{\partial \phi}{\partial y} & \frac{\partial \phi}{\partial z}
\end{pmatrix} 
= 
\begin{pmatrix}
0 & 0 & -\frac{1}{\sqrt{1 - z^2}} \\
-\frac{y}{x^2 + y^2} & \frac{x}{x^2 + y^2} & 0
\end{pmatrix}.
\end{align}
The push-forward of the vector field $(0, z, -y)$, which is $L^x$ under the map $f$ is:
\begin{align}
f_*(0, z, -y) = 
\begin{pmatrix}
\frac{y}{\sqrt{1 - z^2}} \\
\frac{xz}{x^2 + y^2}
\end{pmatrix}.
\end{align}

Now if we put,
\begin{align}
x & = \sin \theta \cos \phi, \\
y & = \sin \theta \sin \phi, \\
z & = \cos \theta,
\end{align} 
we get the killing vector field in the $\partial_{\theta}, \partial_{\phi}$ basis as, $(\sin\phi,\cot\theta\cos\phi)$.

**Question 1** : Method 2 seems to be the flawless way of handling the situation when working with general manifolds, but why do the other methods give the correct answer for the Killing Fields?

**Question 2** : What is the interpretation of the different expressions for $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial y}$ that we get from equations (4),(5) and (7)? In which context are the expressions (4) and (5) meaningful when we are talking about mapping between two manifolds?

Method 2 is standard in quantum mechanics but apparently it seems that there is more to it when we are working with fields on manifolds.